Ricci-Flat Branes

TL;DR

Problem: generalize brane worldvolume geometries to curved, Ricci-flat manifolds and determine when such configurations preserve supersymmetry. Approach: analyze the D8-brane in Romans massive IIA, derive a Ricci-flat worldvolume ansatz with functions F(y) and φ(y), study Killing spinor conditions, and extend the construction to domain walls and magnetic branes across dimensions; classify admissible worldvolume geometries by holonomy and provide explicit non-compact examples. Key results: D8-brane and reductions admit supersymmetric, BPS solutions when the worldvolume is Ricci-flat and supports Killing spinors; explicit Spin(7) and G2 holonomy manifolds realize such worldvolumes; the same Ricci-flat generalization applies to domain walls and magnetic branes, with persistent worldvolume solitons. Significance: broadens the spectrum of supersymmetric brane solutions, links brane geometry to gravitational instantons and holonomy, and suggests connections to string beta-functions and M-theory origins of D8-branes.

Abstract

Up to overall harmonic factors, the D8-brane solution of the massive type IIA supergravity theory is the product of nine-dimensional Minkowski space (the worldvolume) with the real line (the transverse space). We show that the equations of motion allow for the worldvolume metric to be generalised to an arbitrary Ricci-flat one. If this nine-dimensional Ricci-flat manifold admits Killing spinors, then the resulting solutions are supersymmetric and satisfy the usual Bogomol'nyi bound, although they preserve fewer than the usual one half of the supersymmetries. We describe the possible choices of such manifolds, elaborating on the connection between the existence of Killing spinors and the self-duality condition on the curvature two-form. Since the D8-brane is a domain wall in ten dimensions, we are led to consider the general case: domain walls in any supergravity theory. Similar considerations hold here also. Moreover, it is shown that the worldvolume of any magnetic brane --- of which the domain walls are a specific example --- can be generalised in precisely the same way. The general class of supersymmetric solutions have gravitational instantons as their spatial sections. Some mention is made of the worldvolume solitons of such branes.